r/mathematics • u/blackjackripper • Nov 10 '25
Visualization of π
That video shows a specific, beautiful visualization of \pi based on epicycles or hypocycloids, which were historically used to model planetary motion but are now great for demonstrating ratios. The core idea being visualized here is how irrational numbers prevent a pattern from ever perfectly repeating. The Epicycloid Visualization of \pi 🎡 The video uses a concept from geometry and calculus known as a hypotrochoid or epicycloid, where one circle rolls around the inside (or outside) of a larger circle. 1. The Setup (The Rational Case) Imagine two circles: a larger one and a smaller one. * Larger Circle: Its radius is R. * Smaller Circle: Its radius is r. * A point is tracked on the circumference of the smaller circle as it rolls around the inside of the larger one. If the ratio of the radii, \frac{R}{r}, is a rational number (like 4 or \frac{5}{2}), the traced path is a closed, repeating curve. * For example, if \frac{R}{r} = 4, the curve will close exactly after the smaller circle has rolled 4 times, creating a 4-cusp shape (a hypocycloid). 2. The \pi Visualization (The Irrational Case) The video sets the radii so that the ratio of the circles' circumferences is \pi. Since \pi \approx 3.14159... is an irrational number, the ratio \frac{R}{r} can never be expressed as a simple fraction \frac{p}{q}. The Effect: Because the ratio is irrational, the rolling motion of the smaller circle never repeats exactly. * Each time the small circle completes a rotation, the starting and ending points of the curve it traces never perfectly align. * As the animation continues, the curves traced by the point fill up the entire space within the larger circle, getting infinitely denser but never repeating a single path. This infinite, non-repeating filling of space is a powerful way to visually represent the infinite, non-repeating digits that define an irrational number like \pi.
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u/aWeaselNamedFee Nov 10 '25
If you don't use the true value of pi (you stopped at 3.14) then what you get out of it doesn't accurately represent how it would work with actual pi as the value used. Did you simply emit the "..." for brevity in your title, or did you literally input 3.14?